 Research Article
 Open Access
 Published:
Fixed Points and Stability of the Cauchy Functional Equation in Algebras
Fixed Point Theory and Applications volumeÂ 2009, ArticleÂ number:Â 809232 (2009)
Abstract
Using the fixed point method, we prove the generalized HyersUlam stability of homomorphisms in algebras and Lie algebras and of derivations on algebras and Lie algebras for the Cauchy functional equation.
1. Introduction and Preliminaries
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized HyersUlam stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by GÄƒvruÅ£a [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [6â€“19]).

J.
M. Rassias [20, 21] following the spirit of the innovative approach of Th. M. Rassias [4] for the unbounded Cauchy difference proved a similar stability theorem in which he replaced the factor by for with (see also [22] for a number of other new results).
We recall a fundamental result in fixed point theory.
Let be a set. A function is called a generalized metric on if satisfies
(1) if and only if ;
(2) for all ;
(3) for all .
Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
for all nonnegative integers or there exists a positive integer such that
(1);
(2)the sequence converges to a fixed point of ;
(3) is the unique fixed point of in the set ;
(4) for all .
This paper is organized as follows. In Sections 2 and 3, using the fixed point method, we prove the generalized HyersUlam stability of homomorphisms in algebras and of derivations on algebras for the Cauchy functional equation.
In Sections 4 and 5, using the fixed point method, we prove the generalized HyersUlam stability of homomorphisms in Lie algebras and of derivations on Lie algebras for the Cauchy functional equation.
2. Stability of Homomorphisms in Algebras
Throughout this section, assume that is a algebra with norm and that is a algebra with norm .
For a given mapping , we define
for all and all .
Note that a linear mapping is called a homomorphism in algebras if satisfies and for all .
We prove the generalized HyersUlam stability of homomorphisms in algebras for the functional equation .
Theorem 2.1.
Let be a mapping for which there exists a function such that
for all and all . If there exists an such that for all , then there exists a unique algebra homomorphism such that
for all .
Proof.
Consider the set
and introduce the generalized metric on :
It is easy to show that is complete.
Now we consider the linear mapping such that
for all .
By [23, Theorem 3.1],
for all .
Letting and in (2.2), we get
for all . So
for all . Hence .
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of , that is,
for all . The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (2.12) such that there exists satisfying
for all .
(2) as . This implies the equality
for all .
(3), which implies the inequality
This implies that the inequality (2.5) holds.
It follows from (2.2) and (2.15) that
for all . So
for all .
Letting in (2.2), we get
for all and all . By a similar method to above, we get
for all and all . Thus one can show that the mapping is linear.
It follows from (2.3) that
for all . So
for all .
It follows from (2.4) that
for all . So
for all .
Thus is a algebra homomorphism satisfying (2.5), as desired.
Corollary 2.2.
Let and be nonnegative real numbers, and let be a mapping such that
for all and all . Then there exists a unique algebra homomorphism such that
for all .
Proof.
The proof follows from Theorem 2.1 by taking
for all . Then and we get the desired result.
Theorem 2.3.
Let be a mapping for which there exists a function satisfying (2.2), (2.3), and (2.4). If there exists an such that for all , then there exists a unique algebra homomorphism such that
for all .
Proof.
We consider the linear mapping such that
for all .
It follows from (2.10) that
for all . Hence, .
By Theorem 1.1, there exists a mapping such that
(1) is a fixed point of , that is,
for all . The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (2.33) such that there exists satisfying
for all .
(2) as . This implies the equality
for all .
(3), which implies the inequality
which implies that the inequality (2.30) holds.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.
Let and be nonnegative real numbers, and let be a mapping satisfying (2.25), (2.26), and (2.27). Then there exists a unique algebra homomorphism such that
for all .
Proof.
The proof follows from Theorem 2.3 by taking
for all . Then and we get the desired result.
3. Stability of Derivations on Algebras
Throughout this section, assume that is a algebra with norm .
Note that a linear mapping is called a derivation on if satisfies for all .
We prove the generalized HyersUlam stability of derivations on algebras for the functional equation .
Theorem 3.1.
Let be a mapping for which there exists a function such that
for all and all . If there exists an such that for all . Then there exists a unique derivation such that
for all .
Proof.
By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive linear mapping satisfying (3.3). The mapping is given by
for all .
It follows from (3.2) that
for all . So
for all . Thus is a derivation satisfying (3.3).
Corollary 3.2.
Let and be nonnegative real numbers, and let be a mapping such that
for all and all . Then there exists a unique derivation such that
for all .
Proof.
The proof follows from Theorem 3.1 by taking
for all . Then and we get the desired result.
Theorem 3.3.
Let be a mapping for which there exists a function satisfying (3.1) and (3.2). If there exists an such that for all , then there exists a unique derivation such that
for all .
Proof.
The proof is similar to the proofs of Theorems 2.3 and 3.1.
Corollary 3.4.
Let and be nonnegative real numbers, and let be a mapping satisfying (3.7) and (3.8). Then there exists a unique derivation such that
for all .
Proof.
The proof follows from Theorem 3.3 by taking
for all . Then and we get the desired result.
4. Stability of Homomorphisms in Lie Algebras
A algebra , endowed with the Lie product on , is called a Liealgebra (see [9â€“11]).
Definition 4.1.
Let and be Lie algebras. A linear mapping is called aLiealgebra homomorphism if for all .
Throughout this section, assume that is a Lie algebra with norm and that is a algebra with norm .
We prove the generalized HyersUlam stability of homomorphisms in Lie algebras for the functional equation .
Theorem 4.2.
Let be a mapping for which there exists a function satisfying (2.2) such that
for all . If there exists an such that for all , then there exists a unique Lie algebra homomorphism satisfying (2.5).
Proof.
By the same reasoning as the proof of Theorem 2.1, there exists a unique linear mapping satisfying (2.5). The mapping is given by
for all .
It follows from (4.1) that
for all . So
for all .
Thus is a Lie algebra homomorphism satisfying (2.5), as desired.
Corollary 4.3.
Let and be nonnegative real numbers, and let be a mapping satisfying (2.25) such that
for all . Then there exists a unique Lie algebra homomorphism satisfying (2.28).
Proof.
The proof follows from Theorem 4.2 by taking
for all . Then and we get the desired result.
Theorem 4.4.
Let be a mapping for which there exists a function satisfying (2.2) and (4.1). If there exists an such that for all , then there exists a unique Lie algebra homomorphism satisfying (2.30).
Proof.
The proof is similar to the proofs of Theorems 2.3 and 4.2.
Corollary 4.5.
Let and be nonnegative real numbers, and let be a mapping satisfying (2.25) and (4.5). Then there exists a unique Lie algebra homomorphism satisfying (2.38).
Proof.
The proof follows from Theorem 4.4 by taking
for all . Then and we get the desired result.
Definition 4.6.
A algebra , endowed with the Jordan product for all , is called aJordanalgebra (see [25]).
Definition 4.7.
Let and be Jordan algebras.
(i)A linear mapping is called a Jordanalgebra homomorphism if for all .
(ii)A linear mapping is called a Jordan derivation if for all .
Remark 4.8.
If the Lie products in the statements of the theorems in this section are replaced by the Jordan products , then one obtains Jordan algebra homomorphisms instead of Lie algebra homomorphisms.
5. Stability of Lie Derivations on Algebras
Definition 5.1.
Let be a Lie algebra. A linear mapping is called aLie derivation if for all .
Throughout this section, assume that is a Lie algebra with norm .
We prove the generalized HyersUlam stability of derivations on Lie algebras for the functional equation .
Theorem 5.2.
Let be a mapping for which there exists a function satisfying (3.1) such that
for all . If there exists an such that for all . Then there exists a unique Lie derivation satisfying (3.3).
Proof.
By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive linear mapping satisfying (3.3). The mapping is given by
for all .
It follows from (5.1) that
for all . So
for all . Thus is a derivation satisfying (3.3).
Corollary 5.3.
Let and be nonnegative real numbers, and let be a mapping satisfying (3.7) such that
for all . Then there exists a unique Lie derivation satisfying (3.9).
Proof.
The proof follows from Theorem 5.2 by taking
for all . Then and we get the desired result.
Theorem 5.4.
Let be a mapping for which there exists a function satisfying (3.1) and (5.1). If there exists an such that for all , then there exists a unique Lie derivation satisfying (3.11).
Proof.
The proof is similar to the proofs of Theorems 2.3 and 5.2.
Corollary 5.5.
Let and be nonnegative real numbers, and let be a mapping satisfying (3.7) and (5.5). Then there exists a unique Lie derivation satisfying (3.12).
Proof.
The proof follows from Theorem 5.4 by taking
for all . Then and we get the desired result.
Remark 5.6.
If the Lie products in the statements of the theorems in this section are replaced by the Jordan products , then one obtains Jordan derivations instead of Lie derivations.
References
Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941,27(4):222â€“224. 10.1073/pnas.27.4.222
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64â€“66. 10.2969/jmsj/00210064
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297â€“300. 10.1090/S00029939197805073271
GÄƒvruÅ£a P: A generalization of the HyersUlamRassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431â€“436. 10.1006/jmaa.1994.1211
Park CG: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002,275(2):711â€“720. 10.1016/S0022247X(02)003864
Park CG: Modified Trif's functional equations in Banach modules over a algebra and approximate algebra homomorphisms Journal of Mathematical Analysis and Applications 2003,278(1):93â€“108. 10.1016/S0022247X(02)005735
Park CG: On an approximate automorphism on a algebra Proceedings of the American Mathematical Society 2004,132(6):1739â€“1745. 10.1090/S0002993903072526
Park CG: Lie homomorphisms between Lie algebras and Lie derivations on Lie algebras Journal of Mathematical Analysis and Applications 2004,293(2):419â€“434. 10.1016/j.jmaa.2003.10.051
Park CG: Homomorphisms between Lie algebras and CauchyRassias stability of Lie algebra derivations Journal of Lie Theory 2005,15(2):393â€“414.
Park CG: Homomorphisms between Poisson algebras Bulletin of the Brazilian Mathematical Society 2005,36(1):79â€“97. 10.1007/s005740050029z
Park CG: HyersUlamRassias stability of a generalized EulerLagrange type additive mapping and isomorphisms between algebras Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(4):619â€“632.
Park C: Fixed points and HyersUlamRassias stability of CauchyJensen functional equations in Banach algebras. Fixed Point Theory and Applications 2007, 2007:15.
Park C, Cho YS, Han MH: Functional inequalities associated with Jordanvon Neumanntype additive functional equations. Journal of Inequalities and Applications 2007, 2007:13.
Park C, Cui J: Generalized stability of ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:6.
Park CG, Hou J: Homomorphisms between algebras associated with the Trif functional equation and linear derivations on algebras. Journal of the Korean Mathematical Society 2004,41(3):461â€“477.
Rassias ThM: Problem 16; 2, Report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 1990,39(2â€“3):292â€“293, 309.
Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000,246(2):352â€“378. 10.1006/jmaa.2000.6788
Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264â€“284. 10.1006/jmaa.2000.7046
Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982,46(1):126â€“130. 10.1016/00221236(82)900489
Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences MathÃ©matiques 1984,108(4):445â€“446.
Rassias JM: Solution of a problem of Ulam. Journal of Approximation Theory 1989,57(3):268â€“273. 10.1016/00219045(89)900415
CÄƒdariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 4):1â€“7.
Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968,74(2):305â€“309. 10.1090/S000299041968119330
Fleming RJ, Jamison JE: Isometries on Banach Spaces: Function Spaces, Chapman & Hall/CRC Monographs and Surveys in Pure & Applied Mathematics. Volume 129. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2003:x+197.
Acknowledgment
This work was supported by Korea Research Foundation Grant KRF2008313C00041.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Park, C. Fixed Points and Stability of the Cauchy Functional Equation in Algebras. Fixed Point Theory Appl 2009, 809232 (2009). https://doi.org/10.1155/2009/809232
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/809232
Keywords
 Linear Mapping
 Functional Equation
 Unique Mapping
 Stability Problem
 Contractive Mapping